The period-index problem and Hodge theory

Conditional on the Lefschetz standard conjecture in degree 2, we prove that the index of a Brauer class on a smooth projective variety divides a fixed power of its period, uniformly in smooth families. In the other direction, we reinterpret in more classical terms recent work of Hotchkiss which gives Hodge-theoretic lower bounds on the index of Brauer classes. We also prove versions of our results over arbitrary algebraically closed base fields, and as an application construct qualitatively new counterexamples to the integral Tate conjecture.